His family came from Germany and he was the son of a music master. He first became interested in science at the age of 14, when visiting his mineralogist uncle in Hamburg, and studied chemistry at the Royal Polytechnic Institution in London. In 1845 he became one of the twenty-six founding students, under A.W.von Hofmann, of the Royal College of Chemistry. Such was his aptitude for the subject that within two years he became von Hermann's assistant and demonstrator. In 1851 Abel was appointed Lecturer in Chemistry, succeeding Michael Faraday, at the Royal Military Academy, Woolwich, and it was while there that he wrote his Handbook of Chemistry, which was co-authored by his assistant, Charles Bloxam.

Abel's four years at the Royal Military Academy served to foster his interest in explosives, but it was during his thirty-four years, beginning in 1854, as Ordnance Chemist at the Royal Arsenal and at Woolwich that he consolidated and developed his reputation as one of the international leaders in his field. In 1860 he was elected a Fellow of the Royal Society, but it was his studies during the 1870s into the chemical changes that occur during explosions, and which were the subject of numerous papers, that formed the backbone of his work. It was he who established the means of storing gun-cotton without the danger of spontaneous explosion, but he also developed devices (the Abel Open Test and Close Test) for measuring the flashpoint of petroleum. He also became interested in metal alloys, carrying out much useful work on their composition. A further avenue of research occurred in 1881 when he was appointed a member of the Royal Commission set up to investigate safety in mines after the explosion that year in the Sealham Colliery. His resultant study on dangerous dusts did much to further understanding on the use of explosives underground and to improve the safety record of the coal-mining industry. The achievement for which he is most remembered, however, came in 1889, when, in conjunction with Sir James Dewar, he invented cordite. This stable explosive, made of wood fibre, nitric acid and glycerine, had the vital advantage of being a 'smokeless powder', which meant that, unlike the traditional ammunition propellant, gunpowder ('black powder'), the firer's position was not given away when the weapon was discharged. Although much of the preliminary work had been done by the Frenchman Paul Vieille, it was Abel who perfected it, with the result that cordite quickly became the British Army's standard explosive.

Abel married, and was widowed, twice. He had no children, but died heaped in both scientific honours and those from a grateful country.

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